Following [1]} we introduce \(2\textsc {N}\) -component KP (\(\textsc {N}\) -component Toda) tau function in form of a
fermionic expectation value which we choose as follows:
\(\langle 0|e^{\sum _{a}\sum _{k>0}(-1)^{a+1}\frac{1}{k} p_k J^{(a)}_k}e^{\sum _{1\le a\le 2\textsc {N}}^\circlearrowleft \sum _i \psi ^{\dag (a)}_{-i-1}\psi ^{(a+1)}_{i}} |0\rangle =:\tau _o(\mathbf {p}^1,\dots ,\mathbf {p}^\textsc {N})\) 
